Density functional theory metadynamics of silver, caesium
and palladium diffusion at b-SiC grain boundaries
Jeremy Rabone
a,⇑, Eddie López-Honorato
ba
European Commission, Joint Research Centre, Institute for Transuranium Elements, D-76125 Karlsruhe, Germany
b
Centro de Investigación y de Estudios Avanzados del IPN (CINVESTAV), Unidad Saltillo, Industria Metalúrgica 1062, Parque Industrial, Ramos Arizpe 25900, Coahuila, Mexico
h i g h l i g h t s
DFT metadynamics of diffusion of Pd, Ag and Cs on grain boundaries in b-SiC. The calculated diffusion rates for Pd and Ag tally with experimental release rates. A mechanism of release other than grain boundary diffusion seems likely for Cs.
a r t i c l e
i n f o
Article history: Received 22 May 2014 Accepted 9 November 2014 Available online 4 December 2014
a b s t r a c t
The use of silicon carbide in coated nuclear fuel particles relies on this materials impermeability towards fission products under normal operating conditions. Determining the underlying factors that control the rate at which radionuclides such as Silver-110m and Caesium-137 can cross the silicon carbide barrier layers, and at which fission products such as palladium could compromise or otherwise alter the nature of this layer, are of paramount importance for the safety of this fuel. To this end, DFT-based metadynam-ics simulations are applied to the atomic diffusion of silver, caesium and palladium along aR5 grain boundary and to palladium along a carbon-richR3 grain boundary in cubic silicon carbide at 1500 K. For silver, the calculated diffusion coefficients lie in a similar range (7.04 1019–3.69 1017m2s1) as determined experimentally. For caesium, the calculated diffusion rates are very much slower (3.91 1023–2.15 1021m2s1) than found experimentally, suggesting a different mechanism to the simulation. Conversely, the calculated atomic diffusion of palladium is very much faster (7.96 1011–7.26 109m2s1) than the observed penetration rate of palladium nodules. This points to the slow dissolution and rapid regrowth of palladium nodules as a possible ingress mechanism in addi-tion to the previously suggested migraaddi-tion of entire nodules along grain boundaries. The diffusion rate of palladium along theR3 grain boundary was calculated to be slightly slower (2.38 1011–8.24 1010 -m2s1) than along theR5 grain boundary. Rather than diffusing along the precise plane of the boundary, the palladium atom moves through the bulk layer immediately adjacent to the boundary as there is greater freedom to move.
Ó 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).
1. Introduction
Understanding the atomic bases for processes such as diffu-sion provides useful input for the interpretation of experimental measurements. Even if a process cannot be adequately modelled in its entirety, the most common reason being the sheer size and complexity of model that would be required at the atomic scale, atomistic models can highlight the underlying interactions and help to narrow the number of possible interpretations of exper-imental observations. Experexper-imental investigations of silicon
carbide as the barrier layers in TRISO (Tristructural Isotropic) fuel particles have been conducted for over forty years. There are numerous complexities associated with such investigations, implanting guest atoms without significantly altering the nature of the silicon carbide prior to the experiment and controlling the microstructure of the silicon carbide layers are two difficulties. Even when such factors are dealt with, there is the remaining problem that diffusion measurements on such samples poten-tially include contributions from many competing mechanisms. Hence to identify the mechanisms that make the greatest contri-butions and so focus on improving these material properties requires information that can only be obtained through atomistic simulations.
http://dx.doi.org/10.1016/j.jnucmat.2014.11.032 0022-3115/Ó 2014 The Authors. Published by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).
⇑Corresponding author. Tel.: +49 7247 951297. E-mail address:[email protected](J. Rabone).
Contents lists available atScienceDirect
Journal of Nuclear Materials
It seems clear from both experiments[1–4]and simulations[5– 8]that grain boundary diffusion is an important mechanism for diffusion in silicon carbide and that the microstructure[9]of the silicon carbide layer is critical for the design and operation of TRISO particle fuel. The potential for the escape of radionuclides such as
110mAg (a strong
c
emitter) and 137Cs (has high environmentalmobility) through the silicon carbide layer of TRISO is a factor in their use in high-temperature reactors. In particular, Ag diffusion has received special attention since is can apparently escape far more readily than any other element, even in intact fuel, thus making it a radiological hazard. Additionally, the corrosion of the silicon carbide layers of TRISO particles by radiogenic palladium
[10]is also an important consideration in the use of such fuels. Density function theory (DFT) molecular dynamics and metady-namics simulations were previously applied to the high-tempera-ture diffusion of Ag and Cs along the symmetric tilt (120) R5 grain boundary[6]. In the initial stages of these simulations, the two atoms relaxed to two different sites and the dynamic simula-tions were started from these two configurasimula-tions. Static calcula-tions confirmed that the configuracalcula-tions in which the posicalcula-tions of the atoms were exchanged were also feasible. Therefore it is desir-able to investigate the dynamics starting from this second pair of configurations to obtain further sampling of the diffusion energy barriers and to see whether the diffusion trajectories depend strongly on the starting configuration. The diffusion of palladium along theR5 boundary was also investigated for comparison with silver diffusion and because on the basis of static DFT calculations of palladium, silver, tin and caesium with silicon carbide[11], pal-ladium is the more easily accommodated in the silicon carbide structure. The diffusion of palladium along an anti-phase R3 (1 1 1) tilt grain boundary (the carbon rich boundary) was also investigated, to asses the importance of the grain boundary struc-ture and free space on the diffusion rate of palladium.
2. Methodology
The computational methods used for these calculations were identical to those used before[6]. In summary, the calculations were carried out using the QUICKSTEP [12,13] module of CP2K (development version 2.2.214) [14] with a mixed Gaussian and plane waves basis[15]. Periodic boundary conditions were applied in all three dimensions. The spin-polarised PBE functional was used[16] with Goedecker–Teter–Hutter (GTH) pseudopotentials
[17–19]incorporating scalar-relativistic core corrections. The orbi-tal transformation method [20] was employed for an efficient wavefunction optimization. In both carbon and silicon the outer 4 electrons (2s2 2p2/3s2 3p2) were treated as the valence shell
while for palladium the outer 18 electrons (4s24p64d10), for silver
the outer 11 electrons (4d105s1) and for caesium the outer 9
elec-trons (5s25p66s1) were treated as valence. Contracted Gaussian
basis sets of DZVP quality were used with a grid cut off of 200 har-trees[21]. Where required, basis set superposition contributions (BSSE) to the system energy were taken into account using the counterpoise correction method of Boys and Bernardi [22]. The magnitude of the BSSE on the interactions between the Pd, Ag and Cs and the silicon carbide was found to be very small, a few hundredths of an electron volt.
The molecular dynamics simulations were carried out in the Born–Oppenheimer approximation using a timestep of 1 fs with the Always Stable Predictor–Corrector (ASPC) method[23] in a constant pressure and temperature ensemble, thus allowing full flexibility of the simulation cells. Although the use of a constant pressure ensemble increases fluctuations in the instantaneous potential energy of the system compared to a constant volume ensemble, the greater flexibility of the simulation cell better paral-lels the damping of stresses over the extended length scales which
are computationally intractable using DFT. The changes in cell vol-ume and especially cell shape that were observed during the sim-ulations indicated the extreme stresses that would otherwise arise during the simulations. The Canonical Sampling through Velocity Rescaling (CSVR) thermostat-barostat [24] was used with time constant of 10 fs for equilibration and 1000 fs for simulations. The metadynamics [25,26] methods available in CP2 K were applied in exactly the same manner as used previously[6]in order to induce more rapid diffusion of silver, caesium and palladium atoms along the grain boundaries. The collective variables of the metadynamics were the y- and z-coordinates of the palladium, sil-ver or caesium atom in relation to the asil-verage y- and z-coordinates of the silicon and carbon atoms more than 1.9 Å from the plane of the grain boundary. The yz plane is parallel to the plane of the grain boundary and defining the coordinate origin in terms of the coordinates of non-boundary atoms prevents the metadynamics translating the entire simulation cell in space while avoiding noise from the motion of the boundary atoms. Every 5 steps (femtosec-onds) during the metadynamics a new Gaussian was added to
VðtÞ ¼ 0:01X NðtÞ i¼1 exp 1 2 ðyðiÞ yð0ÞÞ2 0:01 ! exp 1 2 ðzðiÞ zð0ÞÞ2 0:01 !!
where V(t) is the history-dependent potential, in hartrees, at time step t and N(t) is the number of accumulated distance variables, y and z, in angstrom. As the metadynamics simulation progressed, this potential gradually forced the diffusing atom over the lower energy barriers into previously unexplored regions of phase space. The sampled energy barriers were then used to calculate diffusion coefficients for the diffusion. Because the instantaneous potential energy of such a small system fluctuates significantly during dynamics and metadynamics, it was necessary to average the potential energy at any given instant in time to obtain meaningful energy barriers. Averaging over 200 fs windows is sufficient to almost remove all of the energy fluctuations from 1500 K simula-tions of the grain boundaries alone and the same window size was used for the metadynamics simulations. The maxima of the standard deviation of the mean at each point in time was used to give an indication of the uncertainty in the energies and hence the uncertainty in the energy barriers. Since this uncertainty dominates the uncertainty in the calculated values for the diffusion coefficients, it is the only source of uncertainty that has been included. In particular the uncertainties from the diffusion step lengths and atomic collision rates, which would be of the order of ±10% of the final diffusion coefficients, are difficult to quantify well from the timescales accessible using these simulations.
Views of the simulation cells for theR5 andR3 grain boundaries are shown inFig. 1, showing the two very different boundary struc-tures. The Cartesian directions shown inFig. 1is the same in all of the following figures. The calculated surface energies of the R5 andR3 grain boundaries were 3.35 and 2.16 J m2 (as there are
two different boundaries inR3, this energy is the average of the two boundaries) respectively and is reflected in the relative abun-dances of CSL grain boundaries in polycrystalline silicon carbide
[27]. An indication of the expected mobility along the boundaries is given by the ‘‘free volume’’, which is the difference between the volume of some sections of the grain boundary cell and the volume that the equivalent number of atoms would occupy as bulk b-SiC. The carbon rich grainR3 boundary occupies less volume than the equivalent amount of bulk silicon carbide with a free volume of 2.0 102mm3m2, while the silicon richR3 boundary has a
free volume of 5.1 102mm3m2. The free volume of the R5 boundary (both boundaries in the cell are identical) is 3.1 102mm3m2.
Note that these volumes are for static calculations at 0 K, and therefore only give a rough guide to relative mobilities since the
freedom of movement of the boundary atoms can also influence diffusion of atoms along the boundary.
3. Results
Figs. 2, 3, 5 and 6 show the metadynamics trajectories and potential energy profiles for the four sets of simulations. It is immediately apparent that the silver atom diffuses relatively slowly, making two transitions in the 3 ps of simulation time, and the palladium diffuses relatively quickly, making 6 transitions in 2 ps in both the R5 and the R3 grain boundaries, while the caesium atom remains in the initial site over the 6 ps simulation.
The diffusion of the silver from site 2 occurs in a similar direc-tion to the diffusion event starting from site 1[6], the motion being primarily along the y-axis. After the initial, rate limiting diffusion step in the y-direction the silver atom continues along the z-axis to rest briefly in the next site. The diffusion in the z-direction occurred unhindered because there was a relatively large void in the structure at that point in the simulation. Several grain bound-ary atoms were therefore able to move around the silver atom as it made this transition.
The caesium atom collides with several of the grain boundary atoms during the metadynamics simulation, knocking them into new positions. This strengthens a cage of atoms surrounding the caesium atom, thereby making diffusion along the grain boundary more difficult. DFT optimised structures from the start and after 6 ps metadynamics are compared inFig. 4. The radial distribution functions centred on the caesium atom are also shown and indicate that the silicon atoms were pushed outwards while the carbon atoms moved inwards. The three closest carbon atoms form weak bonds with the caesium atom that are evidenced by an increase in the Mulliken charges. The cell expanded slightly overall, the volume of the statically optimised cell was 1737 Å3 before the
metadynamics simulation and increased to 1758 Å3afterwards.
The metadynamics of the palladium atom on the R5 grain boundary shows almost continual diffusion of this atom along the boundary. After 1 ps the palladium atom rebounds off a silicon atom and diffuses back in roughly the reverse direction to near where it started. Towards the end of the trajectory and correspond-ing with the final energy barrier at 3730 fs, the Pd atom pushes a
silicon atom ahead of it for a distance of about 4.5 Å, in the process altering the structure of the grain boundary. This would suggest that as Pd diffuses it can modify the characteristics of the grain boundary that in turn could affect, possible increase, the diffusion rate of other elements such as Ag.
On theR3 grain boundary the Pd atom leaves the plane of the boundary in the first 500 fs, crossing the first energy barrier as it does so, and thereafter diffuses parallel to the plane of the bound-ary. The reasons for this behaviour are that the relatively short car-bon–carbon bonds of the grain boundary discourage diffusion in the plane of the boundary while less restricted diffusion pathways are present in the adjacent bulk. Diffusion along the pathways immediately adjacent to the grain boundary is apparently easier than through the bulk and so the palladium atom remains near the grain boundary. The metadynamics trajectory reveals why this is the case – the atoms at the grain boundary are slightly more flexible than those in the bulk and can move further aside as the palladium atom passes (Fig. 7).
The energy barriers jump lengths and approximate collision fre-quencies that were obtained from the simulations are summarised inTable 1.
The uncertainties in the energy barriers inTable 1are the worst-case estimates taken as the maximum standard deviation of the mean for the rolling 200 fs samples over the course of each simula-tion (hence there is a single value for each of the metadynamics sim-ulations conducted). This uncertainty arises from the intrinsic thermal oscillations of the system potential energy and would only be reduced by increasing the cell size. The calculated energy barriers compare well with other DFT calculations of silver[5,7], palladium
[28]and caesium[8]in cubic silicon carbide. The jump distances are estimated from the distances between the start and end positions of the diffusing atoms and are in general slightly longer than the dis-tances between the precise starts and ends of transitions shown in
Figs. 2, 3, 5 and 6since there is a certain degree of unhindered motion of the atoms at these sites. The values for the collision rate are rather approximate because of the few collision events that are sampled and because they are influenced by the metadynamics. This is particularly true of palladium where the collision rate observed in a molecular dynamics simulation (1 ps1) is rather
less than observed in the metadynamics (3 ps1). The former value
gives a better indication of the actual collision rate and has been
(a)
(b)
(c)
x
y
z
X=20.43, Y=10.09, Z=8.54, α=β=γ=90.0° X=19.95, Y=9.82, Z=8.81, α=β=90.0° γ=92.7° X=20.18, Y=10.54, Z=9.56, α=β=γ=90.0°Fig. 1. DFT optimised grain boundary structures (without guest atoms) and dimensions (in Å) for (a) the initial and (b) post MDR5 boundary, compared with (c) theR3 boundary.
(a) (b) (c) 0 2 4 6 8 10 12 14 0 5 10 15 20 z-axis x-axis y-axis 0 5 10 15 20 y-axis 8 9 10 11 12 -21860.0 -21853.8 -21847.5 -21841.3 -21835.0 4000 4500 5000 5500 6000 6500 7000 Potential energy / eV Time / fs
Metadynamics potential energy Ag atom
200 fs average
Fig. 2. The trajectory and potential energy profile of Ag on theR5 grain boundary during metadynamics. (a) and (b) Show the y–z and x–y coordinates of the Ag atom; the blue dot indicates the start location of the atom, and the red dots indicate the start, midpoints, and end of barrier crossings. These points in time are indicated with vertical, black lines on the potential energy curve (c). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
(a)
(b)
(c)
0 1 2 3 4 5 0 2 4 6 8 10 z-axis x-axis y-axis 0 2 4 6 8 10 y-axis 8 9 10 11 12 -21400.0 -21392.5 -21385.0 -21377.5 -21370.0 4000 5000 6000 7000 8000 9000 10000 Potential energy / eV Time / fs Metadynamics potential energyCs atom 200 fs average
Fig. 3. The trajectory and potential energy profile of Cs on theR5 grain boundary during metadynamics. (a) and (b) Show the y–z and x–y coordinates of the Cs atom; the blue dot indicates the start location of the atom. The potential energy curve is shown in (c). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
used in place of the value from the metadynamics simulations, where the reduction in rebounds from the energy barriers has arti-ficially reduced the time between collisions.
Diffusion coefficients were calculated using the sampled energy barriers and treating the diffusion as a two-dimensional random walk[6]. The diffusion coefficients using this model are given by:
D ¼1 2r
2
m
eDE=kTwhere r is the distance the atom travelled between sites when an energy barrier is crossed,DE is the height of the energy barrier, k is Boltzmann’s constant, T the temperature and
m
is the atom(a) (b) 0 1 2 3 4 0 1 2 3 4 2 2.5 3 3.5 4 Radius / Å 2 2.5 3 3.5 4 Radius / Å all C Si all C Si
Fig. 4. DFT optimised structures and associated radial distribution functions around the caesium atom (shown in blue in the left hand diagrams) for the structure before (a) and after (b) the metadynamics simulation. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
(a)
(b)
(c)
0 2 4 6 8 10 -4 -2 0 2 4 6 8 10 z-axis x-axis y-axis -4 -2 0 2y-axis4 6 8 10 8 9 10 11 12 -24310.0 -24302.5 -24295.0 -24287.5 -24280.0 2000 2500 3000 3500 4000 Potential energy / eV Time / fs Metadynamics potential energy Pd atom200 fs average
Fig. 5. The trajectory and potential energy profile of Pd on theR5 grain boundary during metadynamics. (a) and (b) Show the y–z and x–y coordinates of the Pd atom; the blue dot indicates the start location of the atom, and the red dots indicate the start, midpoints, and end of barrier crossings. These points in time are indicated with vertical, black lines on the potential energy curve (c) with the taller lines representing the midpoints. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
collision frequency. As already noted, the uncertainty in the energy barriers dominates the uncertainty in the calculated diffusion coef-ficients and uncertainties from the distances travelled by the atoms and the atom collision frequency, are not included owing the lim-ited number of diffusion events sampled.Fig. 8gives a plot of the diffusion coefficients for each of the sampled energy barriers in each of the metadynamics simulations (the first sampled barriers for Ag and Cs are from the previous calculations[6]). Averaging the med-ian values and the extrema give the following effective diffusion coefficients at 1500 K in m2s1: Pd R3 = 1.4 1010±0.77, Pd R5 = 7.6 1010±0.98, Ag R5 = 5.1 1018±0.86, Cs R5 = 2.9
1022±0.87. As reported previously[6], the calculated diffusion
coef-ficient of Cs for diffusion along theR5 grain boundary is of the same
order as calculated for bulk diffusion[8]and does not explain the measured rates (2.1 1017–2.4 1018m2s1 near 1500 K) [1,2,29]. The further diffusion events sampled for silver increase slightly the uncertainty but shift the median to a slightly higher value (rate calculated from first diffusion event 1.5 1018±0.84),
bringing it closer to the experimentally measured rate (1.6– 7.0 1017m2s1near 1500 K)[4,30].
Experiments of the thinning of silicon carbide by palladium led to a linear relationship between the penetration depth and time
[31], rather than proportionality to the square root of time expected of a simple diffusion process. The observed trend of penetration depth against time gives a palladium reaction rate of 1.11 1012m s1at 1500 K. Comparing this rate with the
calcu-lated atomic diffusion coefficient is somewhat difficult, however the faster atomic diffusion rate of 1.4 1010±0.77m2s1is
sup-ported by experimental observations of palladium travelling well beyond the reaction front and the presence of palladium within silicon carbide without the formation of nodules at the PyC/SiC interface[32,33]. Our work and other reported results[4–6] sup-port the experimental evidence[4] that silver can diffuse by its own depending of the microstructure and without the aid of other elements. However, our results also suggest that as palladium dif-fuses it can modify the characteristics of the grain boundary. This could mean that the rapid diffusion of this element could alter grain boundary structures and lead to higher diffusivities for the otherwise less mobile atoms such as caesium or silver. Further-more, our results show that even for elements as mobile as palla-dium, the characteristics of the grain boundary, or in other words the microstructure, can still influence the diffusion behaviour (palladium diffuses 5 times faster along the R5 grain boundary compared to the R3 boundary). Although recent work in actual irradiated fuel did not show clear evidence of Ag diffusing by the
(a)
(b)
(c)
-4 -2 0 2 4 6 8 -10 -5 0 5 10 z-axis x-axis y-axis -10 -5 y-axis0 5 10 7 8 9 10 11 -28496.5 -28489.3 -28482.2 -28475.1 -28468.0 2000 2500 3000 3500 4000 Potential energy / eV Time / fs Metadynamics potential energy Pd atom200 fs average
Fig. 6. The trajectory and potential energy profile of Pd on theR3 grain boundary during metadynamics. (a) and (b) Show the y–z and x–y coordinates of the Pd atom; the blue dot indicates the start location of the atom, and the red dots indicate the start, midpoints, and end of barrier crossings. These points in time are indicated with vertical, black lines on the potential energy curve (c) with the taller lines representing the midpoints. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 7. Plot of the 2 ps metadynamics trajectory showing the diffusion of the palladium atom parallel to theR3 grain boundary (the boundary plane is shown with a dashed line). The palladium atom is shown in blue, silicon atoms in yellow and carbon atoms in grey. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
formation of a Pd–Ag compound[33]the effect of elements such as palladium or strontium cannot be totally disregarded considering the complexity of the system (temperature, irradiation, presence of other elements, variations in microstructure)[34]. Further work is needed in order to understand how these variables affect the dif-fusion of fission products.
4. Conclusions
At the atomic scale and lengths of time accessible using DFT, diffusion is complicated by the potential for the boundary struc-ture to change during, sometimes with changes induced by the dif-fusing atom itself. Variation of the ground boundary structure and thermal oscillations also influence the nature of the energy barriers that a diffusing atom must pass in order to move, leading to a range of energy barriers and therefore a range of diffusion coefficients. Since the diffusion simulations presented in this manuscript spe-cifically target diffusion of a single atom at a grain boundary, they provide insights into the diffusion mechanisms responsible for dif-fusion rates observed experimentally. Where other mechanisms dominate, for example involving different microstructures or phases, discrepancies between the simulated and experimental diffusion rates are to be expected.
The simulated rate of silver diffusion along theR5 grain bound-ary is in close agreement with the experimentally measured diffu-sion rates. It is therefore highly likely that atomic grain boundary diffusion is the responsible mechanism, since bulk diffusion is cal-culated to be much slower and escape through cracks would be a lot faster. For caesium, the calculated atom diffusion rate is much
slower than the release rates measured in the laboratory. The dif-fusion mechanism of caesium remains to be identified, since the simulated grain boundary diffusion is as energetically unfavour-able as bulk diffusion and neither mechanism explains the experi-mental release of caesium from silicon carbide. What is clear from the simulations is that caesium has an extremely low solubility in bulk silicon carbide as well as a strongly repulsive interaction with the grain boundary. For palladium, our results suggest that not only it can diffuse readily but also that it can modify the character-istics of the grain boundary as it progresses, possibly affecting the diffusion rates of other elements such as Cs and Ag.
To fully model the diffusion processes occurring in the experi-ments would require more sophisticated models that can combine atomic diffusion with segregation rates into different phases as well as modelling the evolution of grain boundary structure over time. The scale of these models precludes the direct use of DFT calculations but the relevant parameters for such models could be obtained from DFT and applied to simulations that would allow the various contributing processes to be identified.
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